Theorem 2omotap 7319

Description: If there is at most one tight apartness on 2_o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotap	⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Proof of Theorem 2omotap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omotaplemst 7318	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
2	1	ex 115	. . . 4 ⊢ (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3		df-stab 832	. . . 4 ⊢ (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
4	2, 3	sylibr 134	. . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → STAB 𝑥 = {∅})
5	4	adantr 276	. 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
6	5	exmid1stab 4237	1 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 831   = wceq 1364  ∃*wmo 2043   ⊆ wss 3153  ∅c0 3446  {csn 3618  EXMIDwem 4223  2_oc2o 6463   TAp wtap 7309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-tr 4128  df-exmid 4224  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-1o 6469  df-2o 6470  df-pap 7308  df-tap 7310

This theorem is referenced by:  exmidmotap  7321